CLASS – 8th CBSE – CUBE AND CUBE ROOTS

WHAT IS A CUBE?

The cube of a number is the number raised to the power 3. Thus,
cube of 2 = 23 = 2 x 2 x 2 = 8,
cube of 5 = 23 = 5 x 5 x 5 = 125.

WHAT IS A PERFECT CUBE?

We know that 23 = 8, 33 = 27, 263 = 216, 73 = 343, 103= 1000.
The numbers 8, 27, 216, 343, 1000, … are called perfect cubes. A natural number is said to be a perfect cube, if it is the cube of some natural number, i.e.,
A natural number n is a perfect cube if there exists a natural number m such that  m X m X m = m3.

HOW TO CHECK WHETHER THE GIVEN NUMBER IS A PERFECT CUBE OR NOT?

Example 1: Show that 729 is a perfect cube.
Solution. Resolving 729 into prime factors, we have
729 =3 x 3 x 3 x 3 x 3 x 3
Here, we find that the prime factor 3 of the given number can be grouped into triplets and no factor is left out. Hence, 729 is a perfect cube.
Also, 729 is the cube of 3 x 3, i.e., 729 = (9)3

Example 2: What is the smallest number by which 1323 may be multiplied so that the product is a perfect cube?
Solution. Resolving 1323 into prime factors, we have
1323 = 3 x 3 x 3 x 7 x 7
Since one more 7 is required to make a triplet of 7, the smallest number by which 1323 should be multiplied to make it a perfect cube is 7.

Example 3: What is the smallest number by which 2375 should be divided so that the quotient may be a perfect cube?
Solution. Resolving 2375 into prime factors, we have
2375 = 5 x 5 x 5 x 19
The factor 5 makes a triplet, and 19 is left out. So, clearly 2375 should be divided by 19 to make it a perfect cube.

PROPERTIES

• If and b are two natural numbers such that a3 =b,  then is called the cube of a.
• If the units digit of a3 is b, then the cubes of all numbers ending with a will have their units digit as b.
• The cubes of all numbers that end in 2 have 8 as the units digit. The cubes of all numbers that end in 3 have 7as the units digit.
• The first odd natural number is the cube of 1. The sum of the next two odd natural numbers is the cube of 2. The sum of the next three odd natural numbers is the cube of 3, and so on.
• The cube rootof a given number is a number, which, when multiplied with itself three times, gives the number.
• If a given number is a perfect cube, then its prime factors will always occur in groups of three.
• The cube root of a number can be found using the prime factorisation method, estimation, successive subtraction and long division methods.